Chapter 4

The Consumer: Marginal Value, Marginal
Utility, and Consumer Surplus

In Chapter 3 we used geometry, in the form of
budget lines and indifference curves, to analyze the behavior of
someone consuming only two goods. In this chapter we redo the
analysis for a consumer buying many goods. We again use geometry, but
in a different way. Each diagram shows on its horizontal axis
quantity of one good, and on its vertical axis something related to
that good (utility, value, marginal utility, marginal value) that
varies with quantity.

We begin in the first part of the chapter by
developing the concepts of marginal utility and marginal value and
showing how they can be used to analyze the behavior of a consumer.
The most important result of that analysis will be that the
consumer's demand curve is identical to his marginal value curve. In
the second part of the chapter that result will be used to derive the
concept of consumer surplus--the answer to the question "How much is
it worth to me to be able to buy some good at a particular price--how
much better off am I than if the good did not exist?" The remainder
of the chapter is a collection of loosely related sections in which I
rederive the equimarginal principle, examine more carefully exactly
what we have been doing in the past two chapters, and use consumer
surplus to analyze the popcorn puzzle discussed in Chapter 2.

MARGINAL UTILITY AND MARGINAL VALUE

So far, we have considered the consumption of only
two goods--simple to graph but hardly realistic. We shall now
consider the more general case of many goods. Since we only have
two-dimensional graph paper, we imagine varying the quantity of one
good while spending whatever income we have left on the optimal
bundle of everything else.

Table 4-1 shows bundles, each of which contains
the same quantity of all goods other than oranges, plus some number
of oranges. In addition to showing the utility of each bundle, it
also shows the marginal
utility for each additional orange--the
increase in utility as a result of adding that orange to the bundle.
Figure 4-1 shows the same information in the form of a graph, with
number of oranges on the horizontal axis and total utility and
marginal utility on the vertical axes. In comparing the table to the
figure, you will note that on the table there is one value of
marginal utility between 1 orange and 2, another between 2 and 3, and
so forth, while on the figure marginal utility changes smoothly with
quantity. The marginal utility shown on the table is really the
average value of marginal utility over the corresponding range. For
example, 20 is the average of marginal utility between 1 and 2
(oranges)--bundles B and C.

Table 4-1

Bundle

Oranges/Week

Total Utility

Marginal Utility

A

0

50

B

1

80

30

C

2

100

20

D

3

115

15

E

4

125

10

F

5

133

8

G

6

139

6

H

7

144

5

I

8

146

2

J

9

147

1

K

10

147

0

L

11

147

0

M

15

147

0

N

20

147

0

Total utility and marginal utility of oranges,
assuming that it costs nothing to dispose of them. Total utility is shown in black, and marginal utility is
shown in color. Because surplus oranges can be freely disposed of,
marginal utility is never negative, and total utility never decreases
with increasing numbers of oranges.

On a table such as Table 4-1, marginal utility is
the difference between the utility of 1 orange and none, between 2
and 1, and so forth. On a graph such as Figure 4-1, it is the slope
of the total utility curve. Both represent the same thing--the rate
at which total utility increases as you increase the quantity of
oranges. Since marginal utility is the slope of total utility, it is
high when total utility is rising steeply, zero when total utility is
constant, and negative if total utility is falling.

Total utility is stated in utiles--hypothetical units of
utility. Since marginal utility is an increase in utility divided by
an increase in oranges, it is measured in utiles per orange. That is
why Figure 4-1 has two different vertical axes, marked off in
different units. Both marginal utility and total utility depend on
quantity of oranges, so both have the same horizontal axis. By
putting them on the same graph, I make it easier to see the
relationship between them.

The idea of total and marginal will be used many
times throughout this book and applied to at least five different
things--utility, value, cost, revenue, and expenditure. In each case,
the relation between total and marginal is the same--marginal is the
slope of total, the rate at which total increases as quantity
increases. In this chapter, we use marginal utility and marginal
value in order to understand consumer choice; in later chapters,
production (both by individuals and by firms) will be analyzed in a
similar way using marginal cost.

Declining Marginal Utility

You are deciding how many oranges to consume. If
the question is whether to have one orange a week or none, you would
much prefer one. If the alternatives are 51 oranges a week or 50, you
may still prefer the additional orange, but the gain to you from one
more orange is less. The marginal utility of an orange to you depends
not only on the orange and you, but also on how many oranges you are
consuming. We would expect the utility to you of a bundle of oranges
to increase more and more slowly with each additional orange. Total
utility increasing more and more slowly means marginal utility
decreasing, as you can see from Table 4-1, so marginal utility
decreases as the quantity of oranges increases. This is what I
earlier called the principle of declining marginal utility. There may
be some point (9 oranges a week on Table 4-1 and Figure 4-1) at which
you have as many oranges as you want. At that point, total utility
stops increasing; additional oranges are no longer a good. Their
marginal utility is zero.

As long as one of the things we can do with
oranges is throw them away, we cannot be worse off having more
oranges; so oranges cannot be a bad. If it were costly to dispose of
oranges (imagine yourself buried in a pile of them), then at some
point the marginal utility of an additional orange would become
negative--you would prefer fewer to more. Figure 4-2 shows your total
and marginal utility for oranges as a function of the quantity of
oranges you are consuming, on the assumption that it is costly to
dispose of oranges.

Total utility and marginal utility of oranges,
assuming that it is costly to dispose of them. I want to eat only 10 oranges, so additional oranges have
negative marginal utility. Total utility falls as the number of
oranges increases beyond 10.

From Marginal Utility to Marginal
Value

Utility is a convenient device for thinking about
choice, but it has one serious limitation--we can never observe it.
We can observe whether bundle X has more utility to you than bundle Y
by seeing which you choose, but that does not tell us how much more utility the bundle
you prefer has. Since utiles are not physical objects that we can
handle, taste, trade, and measure, we can never try the experiment of
offering you a choice between an apple and 3 utiles in order to see
whether the marginal utility of an apple to you is more or less than
3.

What we can observe is the relative marginal
utilities of different goods. If we observe that you prefer 2 apples
to 1 orange, we can conclude that the additional utility you get from
the 2 apples is more than you get from the orange; hence the marginal
utility per apple must be more than half the marginal utility per
orange. If instead of measuring utility in utiles we measure it in
units of the marginal utility of 1 apple, we can then say that the
marginal utility of 1 orange is less than 2. If we observe that you
are indifferent between 3 apples and 1 orange, we can say that the
marginal utility of an orange is exactly 3.

What we are now dealing with is called
marginal value;
it is what one more unit of a good is worth to you in terms of other
goods. Unlike marginal utility, it is in principle (and to some
extent in practice) observable. We cannot watch you choose between
apples and utiles, but we can watch you choose between apples and
oranges. It is what I referred to in the previous chapter as the
value of an orange (measured in apples). A more precise description
would have been "the value of one more orange."

While we could discuss marginal value in terms of
apples, it is easier to discuss it in terms of dollars. "The value to
you of having one more orange is $1" means that you are indifferent
between having one more orange and having one more dollar. Since the
reason we want money is to buy goods with it, that means that you are
indifferent between having one more orange and having whatever goods
you would buy if your income were $1 higher. A graph showing total
and marginal utility (Figure 4-2) and the corresponding graph showing
total and marginal value (Figure 4-3) appear the same, except for the
scale; the vertical axis of one has utiles where the other has
dollars, and $1 need not correspond to one utile. In drawing the
figures, I have assumed that the marginal utility of income is 2
utiles/dollar (an additional $1 is worth 2 utiles), so a marginal
utility of 20 utiles per orange corresponds to a marginal value of
$10/ orange, and a total utility of 60 utiles corresponds to a total
value of $30.

This is an adequate way of looking at the relation
between marginal value and marginal utility so long as we only
consider situations in which the marginal utility of $1 does not
change. If it does, then measuring utility in dollar units is like
measuring a building with a rubber ruler. The resulting problems will
be discussed in the optional section at the end of this chapter. For
the moment, we will assume that the marginal utility of $1 can be
treated as a constant. In that case, marginal value is simply
marginal utility divided by the marginal utility of an additional
dollar of income:

MV(oranges) = MU(oranges)/MU(income).

How Are Marginal Eggs Different From Other
Eggs? You eat some number of eggs each
week. Suppose that the marginal value of the fifth egg (per week) is
$0.50 (per week). This does not mean that there is a particular egg
that is worth $0.50; it means that the difference between having 5
eggs per week and having 4 is worth $0.50/week. If we imagine that 5
eggs per week means 1 egg/day from Monday through Friday (cereal on
the weekend), there is no reason why any one of those eggs should be
valued more than another--but it seems likely that the extra value of
5 eggs a week instead of 4 is less than the extra value of 4 instead
of 3. There is a marginal value of egg, not a marginal egg.

While this is the correct way of looking at
marginal value in general, there are some particular cases in which
one can talk about a marginal
unit--a specific unit that produces the
marginal value. Considering such cases may make it easier to
understand the idea of marginal value. Once understood, it can then
be applied to more general cases.

The Declining Marginal Value of
Water. Suppose, for example, that we use
water for a number of different uses--drinking, washing, flushing,
watering plants, swimming. The value of a gallon of water used in one
way does not depend on how much water we are using in another; each
is independent. To each use we can assign a value per gallon. If the
price of water is $1/gallon, we use it only for those uses where it
is worth at least that much; as the price falls, the number of uses
expands. If water is worth $1/gallon to us for washing but only
$0.10/gallon for swimming and $0.01/gallon for watering the lawn,
then if its price is between $0.10 and $1 we wash but do not swim, if
it is between $0.01 and $0.10 we wash and swim but do not water, and
if it is below $0.01 we do all three. We can then talk of the
marginal use for water--the use that is just barely worthwhile at a
particular price.

Total value and marginal value of oranges.
The marginal utility of income is assumed
to be 2 utiles per dollar, so total value is half as many dollars as
total utility is utiles. The same is true for marginal value and
marginal utility.

If each additional unit of water goes for a
different and independent use, there is an obvious justification for
the principle of declining marginal utility. If you have only a
little water, you use it for the most valuable purposes--drinking,
for example. As you increase your consumption, additional water goes
into less and less important uses, so the benefit to you of each
additional gallon is less than that of the gallon before. In this
particular case, declining marginal utility is not merely something
we observe but also something implied by rationality. The difference
between this and the egg case is that using water for a swimming pool
does not change the value to us of using water to drink or to water
the lawn, whereas eating an egg every Wednesday, in addition to the
Monday, Tuesday, Thursday, and Friday eggs, may make us enjoy the
other four eggs a little less.

The Declining Marginal Value of
Money. Consider, instead of water, money.
There are many different things you can buy with it. Imagine that all
of the things come in $1 packages. You could imagine arranging the
packages in the order of how much you valued them--their utility to
you. If you had $100, you would buy the 100 most valuable packages.
The more money you had, the further down the list of packages you
could go and the less valuable the marginal package would be. So
additional money is worth less to you the more money you have.

While this way of looking at things is useful, it
is not entirely correct, since goods are not independent; the
possession of one may make another more or less valuable. One can
imagine situations in which increasing your income from $3,000/year
to $3,001 was more important than increasing it from $2,000 to
$2,001. You may find it interesting to think up some examples. I will
return to the subject in a later chapter.

Marginal Value and Demand

One of the objectives of this chapter is to derive
a demand curve--a relation between the price of a good and how much
of it a consumer chooses to buy. We are now in a position to do so.
Imagine that you can buy all the eggs you want at a price of
$0.80/egg. You first consider whether to buy 1 egg per week or none.
If the marginal value to you of the first egg is more than $0.80 (in
other words, if you prefer having one more egg to whatever else you
could buy with $0.80), you are better off buying at least 1 egg. The
next question is whether to buy 2 eggs or 1. Again, if the marginal
value of one more egg is greater than $0.80, you are better off
buying the egg and giving up the money. Following out the argument to
its logical end, you conclude that you want to consume eggs at a rate
such that the marginal value of an egg is $0.80. If you increased
your consumption above that point, you would be paying $0.80 for an
additional egg when consuming one more egg per week was worth less
than $0.80 to you (remember declining marginal utility). You would be
consuming an egg that was worth less than it cost. If you consumed
less than that amount, you would fail to consume an egg that was
worth more to you than it cost. This implies that (if you act
rationally) the same points describe both your marginal value for
eggs (value of having one more egg as a function of how many eggs per
week you are consuming) and your demand for eggs (number of eggs per
week you consume as a function of the price of eggs), since at any
price you consume that quantity for which your marginal value of eggs
equals that price. The relation is shown in Figures 4-4a and 4-4b.
Note that your marginal value for eggs shows value per egg as a
function of quantity. Your demand curve shows quantity as a function
of price.

Figures 4-4c and 4-4d show the same relation for a
continuous good. As long as you are consuming a quantity of wine for
which the marginal value of additional wine is greater than its
price, you can make yourself better off by increasing your
consumption. So you buy that quantity for which marginal value equals
price. Since you do that for any price, your demand curve and your
marginal value curve are the same.

By the principle of declining marginal utility,
the marginal value curve should slope down; the more we have, the
less we value additional quantities. I have just demonstrated that
the demand curve is identical to the marginal value curve. It follows
that demand curves slope down.

Marginal value and points on the demand cruve.
Panels (a) and (b) show a lumpy good. At
any price, you buy a quantity for which marginal value equals the
price, so the (price, quantity) points on the demand curve are the
same as the (marginal value, quantity) points on the marginal value
curve. Panels (c) and )d) show a continuous good. At any price, you
buy a quantity for which marginal value equals the price; that is
true for every price, so the demand curve is identical to the
marginal value curve.

Some Problems. There
is one flaw in this argument. So far, I have been assuming that the
marginal utility of income--the increased utility from the goods
bought with an extra dollar--is constant. But just as the marginal
utility of apples depends on how many apples we have, the marginal
utility of income depends on how much income we have. If our income
increases, we will increase the quantities we consume (for normal
goods), reducing the marginal utility of those goods. The marginal
utility of a dollar is simply the utility of the additional goods we
could buy with that dollar; so as income rises, the marginal utility
of income falls.

A marginal value curve shows us what happens when
we increase our consumption of one good while holding everything else constant. This does not quite correspond to what is shown by the
demand curve of Figure 4-4d. That curve graphs quantity against
price. As the price of the good falls and the quantity consumed
increases, the total amount spent on that good changes--and so does
the amount left to spend on other goods. Since the marginal value
curve shows the value of a good measured in money, it should shift
slightly as the change in that good's price changes the amount we
have left to spend on other goods, and hence the marginal utility of
money.

A similar difficulty in the analysis arises when
the value to us of one good depends on how much we have of some other
good. Bread is more valuable when we have plenty of butter, and
butter less valuable when we have plenty of margarine. As price falls
and quantity consumed rises in Figures 4-4b and 4-4d, the quantities
of other goods consumed changes--which may affect the value of the
good whose price has changed.

The problems here are the same as in the case of
the Giffen good discussed earlier; a change in the price of one good
affects not only the cost of that good in terms of others but also
the consumer's total command over goods and services--a drop in price
is equivalent to an increase in income. A full discussion of this
would involve the income-compensated (Hicksian) demand curve
discussed in the previous chapter.

A simpler solution, adequate for most practical
purposes, is the one we used to justify the downward-sloping demand
curve in the previous chapter. Since consumption is usually divided
among many different goods, with only a small part of our income
spent on any one, a change in the price of one good has only a very
small effect on our real income and our consumption of other goods as
compared to its effect on the cost of the good whose price has
changed. If we ignore the small income effect, the complications of
the last few paragraphs disappear. The demand curve is then exactly
the same as the marginal value curve; since the latter slopes down
(because of diminishing marginal utility), so does the former. The
indifference curve argument gave us a downward-sloping demand curve
for a consumer choosing between two goods; this argument gives one in
the general case of a consumer buying many goods.

Warning. When I ask
students taking an exam or quiz to explain why the demand curve is
the same as the marginal value curve, most of them think they know
the answer--and most of them are wrong. The problem seems to be a
confusion based on an imprecise verbal argument. It sounds very
simple: "Your demand is how much you demand something, which is the
same as how much you value it" or, alternatively, "Your demand is how
much you are willing to pay for it, which is how much you value it."
But both of those explanations are wrong. Your demand curve shows not
how much you demand it but how much of it you demand--a quantity, not
an intensity of feeling.

Your demand curve does not show how much you are
willing to give for the good. On Figure 4-4d, the point X (price =
$25/gallon, quantity = 2 gallons/week) is above your demand curve.
But if you had to choose between buying 2 gallons of wine a week at a
price of $25/gallon or buying no wine at all, you would buy the wine;
as we will see in a few pages, its total value is more than its cost.
The demand curve shows the quantity you would choose to buy at any price,
given that (at that price) you were free to buy as much or as little
as you chose. It does not show the highest price you would pay for
any quantity if you were choosing between that quantity and
nothing.

What the height of your demand curve at any price
is equal to is the amount you would be willing to pay for a little
more of the good--your marginal value. That is true--but not because
demand and value mean the same thing. The reason was given in the
discussion of eggs and wine a few paragraphs earlier. It is also
important; as you will see later in the chapter, the relation between
demand and marginal value is essential in deriving consumer surplus,
and as you will see later in the book, consumer surplus is an
important tool in much of economics. I have emphasized the
relationship between the two curves so strongly because it is easy to
skip over it as obvious and continue building the structure of
economics with one of its foundations missing.

Price, Value, Diamonds, and Water

In addition to the downward-sloping demand curve,
another interesting result follows from the analysis of marginal
value. As I pointed out earlier, there is no obvious relation between
price (what you must give up to get something) and value (how much it
is worth to you--what you are willing, if necessary, to give up to
get it), a point nicely summarized in the saying that the best things
in life are free. But if you are able to buy as much as you like of
something at a per-unit price of P, you will choose, for the reasons
discussed above, to consume that quantity such that an additional
unit is worth exactly P to you. Hence in equilibrium (when you are
dividing your income among different goods in the way that maximizes
your welfare), the marginal value of goods is just equal to their
price! If the best things in life really are free, in the sense of
being things of which you can consume as much as you want without
giving up anything else (true of air, not true of love), then their
marginal value
is zero!

This brings us back to the "diamond-water
paradox." Water is far more useful than diamonds, and far cheaper.
The resolution of the paradox is that the total value to us of water
is much greater than the total value of diamonds (we would be worse
off with diamonds and no water than with water and no diamonds), but
the marginal value of water is much less than that of diamonds. Since
water is available at a low cost, we use it for all its valuable
uses; if we used a little more, we would be adding a not very
valuable use, such as watering the lawn once more just in case we had
not watered it quite enough. Diamonds, being rare, get used only for
their (few) valuable uses. Relative price equals relative marginal
value; diamonds are much more expensive than water.

CONSUMER SURPLUS

This brings us to another (and related) paradox.
Suppose you argued that "since the value of everything is equal to
its price, I am no better off buying things than not buying, so I
would be just as happy on Robinson Crusoe's island with nothing for
sale as I am now." You would be confusing marginal value and average
value; you are no better off buying the last drop of water at exactly
its value but are far better off buying (at the same price) all the
preceding (and to you more valuable) drops. Note that "preceding"
describes order in value, not in time.

Can we make this argument more precise? Is there
some sense in which we can define how much better off you are by
being able to buy as much water as you want at $0.01/gallon or as
many eggs as you want at $0.80/egg? The answer is shown in Figure
4-5a. By buying one egg instead of none, you receive a marginal value
of $1.20 and give up $0.80; you are better off by $0.40. Buying a
second egg provides a further increase in value of $1.10 at a cost of
another $0.80. So buying 2 eggs instead of none makes you better off
by $0.70.

This does not mean you have $0.70 more than if you
bought no eggs--on the contrary, you have $1.60 less. It means that
buying 2 eggs instead of none makes you as much better off as would
the extra goods you would buy if your income were $0.70 higher than
it is. You are indifferent between having your present income and
buying 2 eggs (as well as whatever else you would buy with the
income) and having $0.70 more but being forbidden to buy any
eggs.

Continuing the explanation of Figure 4-5a, we see
that as long as you are consuming fewer than 5 eggs per week, each
additional egg you buy makes you better off. When your consumption
reaches 5 eggs per week, any further increase involves buying goods
that cost more than they are worth. The total gain to you from
consuming 5 eggs at a price of $0.80 each instead of consuming no
eggs at all is the sum of the little rectangles shown in the figure.
The first rectangle is a gain of $0.40/egg times 1 egg, for a total
gain of $0.40; the next is $0.30/egg times 1 egg, and so on.

Marginal value curve and consumer surplus for a
lumpy good. The shaded area under the
marginal value curve and above the price equals the benefit to you of
buying that quantity at that price. It is called consumer surplus.

Summing the area of the rectangles may seem odd to
you. Why not simply sum their heights, which represent the gain per
egg at each stage? But consider Figure 4-5b, which shows a marginal
value curve for which the rectangles no longer all have a width of 1
egg per week. Gaining $0.40/egg on 3 eggs is worth 3 times as much as
gaining $0.40/egg on 1 egg.

Finally, consider Figure 4-6a, where instead of a
lumpy good such as eggs we show a continuous good such as wine (or
apple juice). If we add up the gain on buying wine, drop by drop, the
tiny rectangles exactly fill the shaded region A. That is your net
gain from being able to buy wine at $8/gallon.

This area--representing the gain to a consumer
from what he consumes--has a name. It is called consumer surplus. It equals the
area under the demand curve and above the price-area A on Figure
4-6a. You will meet consumer surplus again--its derivation was one of
the main purposes of this chapter. Its traditional use in economics
is to evaluate the net effect on consumers of some change in the
economic system, such as the introduction of a tax or a subsidy. As
we will see in Chapters 10 and 16, it is also sometimes useful for
helping a firm decide how to price its product.

Your consumer surplus from buying wine at some
price is the value to you of being able to buy as much wine as you
wish at that price--the difference between what you pay for the wine
and what it is worth to you. The same analysis can be used to measure
the value to you of other opportunities. Suppose, for example, that
you are simply given 2 gallons per week for free, with no opportunity
to either sell any of it or buy any more. The value to you of what
you are getting is the value of the first drop of wine, plus the
second, plus ... adding up to the whole area under your demand
curve--region A plus region B on Figure 4-6a. The situation is just
the same as if you bought 2 gallons per week at a price of $8/gallon
and were then given back the money. Area A is the consumer surplus on
buying the wine; area B is the $16/week you spend to get it. The
total value to you of the wine is the sum of the two, which is the
area under the marginal value curve; total value is simply the area
under marginal value.

Marginal value and consumer surplus for a
continuouos good. A is the consumer
surplus from beijng able to buy all the wine you want at $8/gallon. B
is what you pay for it. A+B is the total value to you of 2 gallons
per week of wine. B+E+D is what you would pay if you bought 2 gallons
per week at $25/gallon.

If area A plus area B is the value to you of being
given 2 gallons of wine per week, it is also the largest sum you
would pay for 2 gallons per week if the alternative were having no
wine at all. Figure 4-6b shows that situation. Your surplus from
buying 2 gallons per week for $25/gallon is the value to you of the
wine--areas A plus B on the previous figure, equal to C + E + B on
Figure 4-6b--minus what you spend for it. You are spending $25/gallon
and buying 2 gallons, so that comes to $50/week--the colored
rectangle D + E + B on Figure 4-6b. Subtracting that from the value
of the wine (C + E + B) gives a surplus equal to region C minus
region D. Your surplus is positive, so you buy the wine. This is the
case mentioned earlier in the chapter where you would rather have a
price/quantity combination that is above your demand curve than have
nothing.

ODDS AND ENDS

Again the Equimarginal Principle

You are consuming a variety of goods; being
rational, you have adjusted the amount you consume of each until you
are consuming that bundle you prefer among all those bundles you can
buy with your income. Consider two goods--apples and cookies. For
each, consider the marginal utility to you of an additional dollar's
worth of the good. Suppose it were larger for apples than for
cookies. In that case, by spending $1 less on cookies and $1 more on
apples, you could get a better bundle for the same amount of money!
But you are supposed to have already chosen the best possible bundle.
If so, no further change can improve your situation. It follows that
when you have your optimal bundle, the utility to you of a dollar's
worth of apples must be the same as the utility to you of a dollar's
worth of cookies--or a dollar's worth of anything else. If it were
not, there would be a better bundle with the same price, so the one
you had would not be optimal.

Since that may seem confusing, I will go through
it again with numbers. We start by assuming that you are consuming
your optimal bundle of apples and cookies. Suppose apples cost $0.50
each and cookies (the giant size) cost $1 each. You are consuming 4
cookies and 9 apples each week; at that level of consumption, the
marginal utility of a cookie is 3 utiles and the marginal utility of
an apple is 2 utiles (remember that the marginal utility of something
depends both on your preferences and on how much you are consuming).
A dollar's worth of apples is 2 apples; a dollar's worth of cookies
is 1 cookie. If you increased your consumption of apples by 2, your
utility would increase by four utiles; if you then decreased your
consumption of cookies by 1, your utility would go back down by 3
utiles. The net effect would be to make you better off by 1 utile (4
- 3 = 1). You would still be spending the same amount of money on
apples and cookies, so you would have the same amount as before to
spend on everything else. You would be better off than before with
regard to apples and cookies and as well off with regard to
everything else. But that is impossible; since you were already
choosing the optimal bundle, no change in what you consume can make
you better off.

I have proved that if the marginal utility per
dollar's worth of the different goods you are consuming is not the
same, you must not be choosing the optimal bundle. So if you are
choosing the optimal bundle, the marginal utility of a dollar's worth
of any of the goods you consume must be the same. In other words, the
marginal utility of each good must be proportional to its price. If
butter costs $4/pound and gasoline $2/gallon, and a dollar's worth of
butter (1/4 pound) increases your utility by the same amount as a
dollar's worth of gasoline (1/2 gallon), the marginal utility of
butter (per pound) must be twice the marginal utility of gasoline
(per gallon)--just as the price of butter (per pound) is twice the
price of gasoline (per gallon).

This is now the fourth time I have derived this
result. The third was when, in the process of showing that the
marginal value curve and the demand curve are the same, I
demonstrated that you consume any good up to the point where its
marginal value is equal to its price. While I did not point out then
that marginal value equal to price implies the equimarginal
principle, it is easy enough to see that it does. Simply repeat the
argument for every good you consume. If marginal value is equal to
price for every good, then for any two goods, the ratio of their
marginal values is the same as the ratio of their prices. Since
marginal value is marginal utility divided by the marginal utility of
income, the ratio of the marginal values of two goods is the same as
the ratio of their marginal utilities.

This may be clearer if it is stated using algebra
instead of English. Consider two goods X and Y, with marginal values
MVx , and
MVy ,
marginal utilities MUx and MUy , and prices Px and Py . We have

MVx = Px;

MVy = Py;

MVx [[equivalence]] MUx/MU(income);

MVy [[equivalence]] MUy/MU(income).

Therefore,

Px/Py = MVx/MVy = MUx/MUy.

The left hand side of this equation corresponds to
"the price of an apple measured in oranges" in Chapter 3 (minus the
slope of the budget line; apples are X, oranges Y); the right hand
side is the marginal rate of substitution (minus the slope of the
indifference curve).

This is the final derivation of the principle in
this chapter, but you will find it turning up again in economics (and
elsewhere). The form in which we have derived it this time makes more
obvious the reason for calling it the equimarginal principle. A
convenient, if sloppy, misstatement of it is "Everything is equal on
the margin."

It is important, in this and other applications of
the equimarginal principle, to realize that it is a statement not
about the initial situation (preferences, market prices, roads,
checkout counters, or whatever) but about the result of rational
decision. You may (as I do) vastly prefer Kroger chocolate chip
cookies (the kind they used to bake in the store and sell in the deli
section) to apples; if so, you may buy many more cookies than apples.
What the equimarginal principle tells you is that you will buy just
enough more cookies to reduce the marginal utility per dollar of
cookies to that of apples.

Continuous Cookies

It may occur to some of you that there is a
problem with the most recent argument by which I "proved" the
equimarginal principle. I originally defined the marginal utility of
something of which I have n units as the utility of n + 1 units minus
the utility of n units; since marginal value is derived from marginal
utility, it would be defined similarly. Applying this to my example
of 9 apples and 4 cookies, the marginal value of an apple involves
the difference between 9 and 10 and the marginal value of a cookie
involves the difference between 4 and 5. But the change that I
considered involved increasing the consumption of apples from 9 not
to 10 but to 11, and decreasing the consumption of cookies from 4 to
3. Unless the marginal value of the eleventh apple is the same as
that of the tenth (which it should not be, by our assumption of
declining marginal utility) and the marginal value of the fourth
cookie the same as that of the fifth (ditto), the argument as I gave
it is wrong!

The answer to this objection is that although I
have described the marginal utility of an apple or an orange as the
difference between the utility of 10 and the utility of 9, that is
only an approximation. Strictly speaking, we should think of all
goods as consumed in continuously varying quantities (if this
suggests applesauce and cookie crumbs, wait for the discussion of
time in the next section). We should define the marginal utility as
the increased utility from consuming a tiny bit more, divided by the
amount of that tiny bit (and similarly for marginal value). Marginal
value is then the slope of the graph of total value; in Figure 4-7 it
is V/Q.
If, when we are consuming 100 gallons of water per week, an
additional drop (a millionth of a gallon) is worth one
hundred-thousandth of a cent, then the marginal value of water is
.00001 cents/.000001 gallons, which comes out to $0.10/gallon. The
argument of the previous section can then be restated in terms of an
increase in consumption of .002 apples and a decrease in consumption
of .001 cookies. Since we do not expect the marginal value of cookies
to change very much between 4 cookies and 3.999 cookies, the argument
goes through.

The precise definitions of marginal utility (see
the optional section of Chapter 3) and marginal value require
calculus--the marginal value of apples is the derivative of total
value with respect to quantity. Since I am not assuming that all of
my readers know calculus, I use the sort of imprecise language given
above. Precisely the same calculus concept (a derivative) is implicit
in such familiar ideas as speed and acceleration. You might
carelessly say that, having driven 50 miles in an hour, your speed
was 50 miles per hour--but you know that speed is actually an
instantaneous concept and that 50 miles per hour is only an average
(part of the time you were standing still at a stop light, part of it
going at 50, part of it at 65). A precise definition of speed must be
given in terms of small changes in distance divided by the small
amounts of time during which they occur, just as a precise definition
of marginal value is given in terms of small changes in value divided
by the small changes in quantity that cause them.

Economics and Time

In talking or writing about economics, it is often
convenient to describe consumption in terms of quantities--numbers of
apples, gallons of water, and so forth. But 100 apples consumed in a
day are not of the same value to me as 100 apples consumed in a year.
The easiest way to deal with this problem is to think of consumption
in terms of rates instead of quantities--6 apples per week, 7 eggs
per week, and so on. Income is not a number of dollars but rather a
number of dollars per week. Value is also a flow--6 apples per week
are worth, not $3, but $3/week.

If we think of all quantities as flows and limit
ourselves to analyzing situations in which income, prices, and
preferences remain the same for long periods, we avoid most of the
complications that time adds to economics. Many of these
complications are important to understanding the nonstatic world we
live in. But in solving a hard problem, it is often wise to solve the
easier parts first; so in this section of the book, the problems
associated with change are mostly ignored. Once we have a clearly
worked-out picture of static economics, we can use it to understand
more complicated situations--and will, starting in Chapter 12. Until
then, we are doing economics in a perfectly static and predictable
world, in which tomorrow is always like today and next year is always
like this year. That is why, in drawing indifference curve diagrams,
we never considered the possibility that the consumer would spend
only part of his income in order to save the rest for a rainy day;
either it is raining today or there are no rainy days.

Total value and its slope. V/Q is
the average slope of total value between A and B. As V and Q
become very small, A and B move together, and V/Q approaches
the slope of total value at a point-which is marginal value.

Problems associated with time and change are not
the only complications ignored at this point; you might find it
interesting to make a list as we go along, and see how many get dealt
with by the end of the book.

One advantage to thinking of consumption in eggs
per year instead of just eggs is that it lets us vary consumption
continuously. There are severe practical difficulties with changing
the number of eggs you consume by 1/10 of an egg at a time--what do
you do with the rest of it? But it is easy enough to increase the
rate at which you consume eggs by 1/10 of an egg per week--eat, on
average, 5 more eggs per year. Thus lumpy goods become
continuous--and the consumption of continuous goods is, for
mathematical reasons, easier to analyze than the consumption of lumpy
goods. We can then define marginal utility and marginal value in
terms of very small amounts of apples and cookies without first
converting the apples into applesauce and the cookies into a pile of
crumbs.

There is a second problem associated with time
that we should also note. In describing the process of choice, I talk
about "doing this, then doing that, then . . . " For example, I talk
about increasing consumption from 4 apples to 5, then from 5 to 6,
then from . . . and so on. It sounds as though the process happens
over time, but that is deceptive. We are really describing not a
process of consumption going on out in the real world but rather
something happening inside your head--the process of solving the
problem of how much of each good to consume. A more precise
description would be "First you imagine that you choose to consume no
apples and consider the resulting bundle of goods. Then you imagine
that you consume 1 apple instead of none and compare that bundle with
the previous one. Then 2 instead of 1. Then . . . . Finally, after
you have figured out what level of consumption maximizes your
utility, we turn a switch, the game of life starts, and you put your
solution into practice."

If you find it difficult to distinguish time in
the sense of an imaginary series of calculations by which you decide
what to do from the time in which you actually do it, you may instead
imagine, as suggested before, that we are considering a situation
(income, preferences, prices) that will be stable for a long time. We
start by spending a few days experimenting with different consumption
bundles to see which we prefer. The loss from consuming wrong bundles
during the experiment can be ignored, since it is such a short period
compared to the long time during which the solution is put into
practice.

Money, Value, and Prices

Although prices and values are often given in
terms of money, money has nothing essential to do with the analysis.
In demonstrating the equimarginal principle, for example, I converted
cookies into money (bought one less cookie, leaving me with an extra
dollar to spend on something else) and then converted the money into
apples (bought 2 apples for $1). The argument would have been exactly
the same if there were no such thing as money and a cookie simply
exchanged for 2 apples.

We are used to stating prices in money, but prices
can be stated in anything of value. We could define all our prices as
apple prices. The apple price of a cookie, in my example, is 2
apples--that is what you must give up to get a cookie. The apple
price of an apple is 1 (apple). Once you have the price of everything
in terms of apples, you also have the price of everything in terms of
any other good. If a peach exchanges for 4 apples, and 4 apples
exchange for 8 cookies, then the cookie price of a peach is 8.

There are two ways of seeing why this is true. The
simpler is to observe that someone who has cookies and wants peaches
will never pay more than 8 cookies for a peach, since he could always
trade 8 cookies for 4 apples and then exchange the 4 apples for a
peach. Someone who has a peach and wants cookies will never accept
fewer than 8 cookies for his peach, since he could always trade it
for 4 apples and then trade the 4 apples for 8 cookies. If nobody who
is buying peaches will pay more than 8 cookies and nobody selling
them will accept less, the price of a peach (in cookies) must be 8.
The same analysis applies to any other good. So once we know the
price of all goods in terms of one (in this example apples), we can
calculate the price of each good in terms of any other.

This argument depends on an assumption that has so
far been implicit in our analysis--that we can ignore all costs of
buying and selling other than the price paid. This assumption,
sometimes called zero transaction
costs, is a reasonable approximation for
much of our economic activity and one that will be retained through
most of the book. Exceptions are discussed in parts of Chapters 6 and
18. It is not clear that the assumption is reasonable here. Imagine,
for example, that you have 20 automobiles and want a house. The
cookie price of an automobile is 40,000; the cookie price of a house
is 800,000. It seems, from the discussion of the previous paragraph,
that all you have to do to get your house is trade automobiles for
cookies and then cookies for the house.

But where will you put 800,000 cookies while you
wait for the seller of the house to come collect them? How long will
it take you to count them out to him? What condition will the cookies
be in by the time you finish? Clearly, in the real world, there are
some problems with such indirect transactions.

This brings us to the second reason why relative
prices--prices of goods in terms of other goods--must fit the pattern
I have described. Trading huge quantities of apples, cookies,
peaches, or whatever may be very costly for you and me. It is far
less costly for those in the business of such trading--people who
routinely buy and sell carload lots of apples, wheat, pork bellies,
and many other outlandish things and who make their exchanges not by
physically moving the goods around but merely by changing the pieces
of paper saying who owns what, while the goods sit still. For such
professional traders, the assumption of zero transaction costs is
close to being correct. And such traders, in the process of making
their living, force relative prices into the same pattern as would
consumers with zero transaction costs--even if they never consume any
of the goods themselves.

To see how this works, imagine that we start with
a different structure of relative prices. A peach trades for 2 apples
and an apple trades for 4 cookies, but the price of a peach in
cookies is 10. A professional trader in the peach-cookie-apple market
appears. He starts with 10,000 peaches. He trades them for 100,000
cookies (the price of a peach is 10 cookies), buys 25,000 apples with
the 100,000 cookies (the price of an apple is 4 cookies), trades the
apples for 12,500 peaches (the price of a peach in apples is 2). He
has started with 10,000 peaches, shuffled some pieces of paper
representing ownership of peaches, apples, and cookies, and ended up
with 2,500 peaches more than he started with--which he can now
exchange for whatever goods he wants! By repeating the cycle again
and again, he can end up with as many peaches--and exchange them for
as much of anything else--as he wants.

So far, I have assumed that such a
transaction--the technical name for it is arbitrage--has no effect on the
relative prices of the goods traded. But if you can get peaches, in
effect, for nothing, simply by shuffling a few pieces of paper
around, there is an almost unlimited number of people willing to do
it. When the number of traders--or the quantities each
trades--becomes large enough, the effect is to change relative
prices. Everyone is trying to sell peaches for cookies at a price of
10 cookies for a peach. The result is to drive down the price of
peaches measured in cookies--the number of cookies you can get for a
peach. Everyone is trying to buy apples with cookies at 4 cookies for
an apple. The result is to drive up the price of apples measured in
cookies and, similarly, to drive up the price of peaches measured in
apples. As prices change in this way, the profit from arbitrage
becomes smaller and smaller. If the traders have no transaction costs
at all, the process continues until there is no profit. When that
point is reached, relative prices exactly fit the pattern described
above--you get the same number of cookies for your peach whether you
trade directly or indirectly via apples. If the traders have some
transaction costs, the result is almost the same but not quite;
discrepancies in relative prices can remain as long as they are small
enough so that it does not pay traders to engage in the arbitrage
trades that would eliminate them.

I have now shown that the price of peaches in
terms of cookies is determined once we know the price of both goods
in apples--precisely, if transaction costs are zero; approximately,
if they are not. By similar arguments, we could get the exchange
ratio between any two goods (how many of one must you give for one of
the other) starting with the price of both of them in apples, or in
potatoes, or in anything else. The equimarginal principle then
appears as "the ratio of marginal utilities of two goods is the same
as their exchange ratio." If 2 apples exchange for 1 cookie, then in
equilibrium a cookie must have twice the marginal utility of an
apple.

I used money in talking about values as well as in
talking about prices. Here too, the money is merely a convenient
expository device. The statement that the marginal value of something
is $0.80 means that you are indifferent between one more unit of it
and whatever else you would buy if you had an additional $0.80. Just
as in the case of prices, the money serves as a conceptual
intermediate--we are really comparing one consumption good with
another. The arguments of this chapter could be made in "potato
values" just as easily as in "dollar values." Indeed potato values
are more fundamental than dollar values, as you can easily check by
having a hamburger and a plate of french-fried dollars for
lunch.

It is often asserted that economics is about money
or that what is wrong with economics is that it only takes money into
account. That is almost the opposite of the truth. While money does
play an important role in a few areas of economics such as the
analysis of business cycles, price theory could be derived and
explained in a pure barter economy without ever mentioning
money.

A similar error is the idea that economists assume
everyone wishes to maximize his wealth or his income. Such an
assumption would be absurd. If you wished to maximize your wealth,
you would never spend any money except for things (such as food) that
you required in order to earn more money. If you wished to maximize
your income, you would take no leisure (except that needed for your
health) and always choose the highest paying job, independent of how
pleasant it was. What we almost always do assume is that everyone
prefers more wealth to less and more income to less, everything else
held constant. To say that you would like a raise is not the same
thing as to say that you would like it whatever its cost in
additional work.

Conclusion: Consumption, Languages, and
All That

In my analysis of consumption (Chapters 3 and 4),
I have tried to do two things. The first is to show how rational
behavior may be analyzed in a number of different ways, each
presenting the same logical structure in a different language. The
second is to use the analysis to derive three interrelated
results.

The simplest of the three, derived once with
indifference curves and once with marginal value, is that demand
curves slope down--the lower the price of something, the more you
buy. In both cases, the argument depends on declining marginal
utility. In both cases, there is a possible exception, based on the
ambiguity between a fall in price and a rise in income; in both
cases, the ambiguity vanishes if we insist on a pure price change--a
change in one price balanced by either a change in the other
direction of all other prices or a corresponding change in income. It
also vanishes if we assume that any one good makes up a small enough
part of our consumption that we may safely ignore the effect on our
real income of a change in its price.

A second result is that the value to a consumer of
being able to buy a good at a price, which we call consumer surplus,
equals the area under the demand curve and above the price. At this
point, that may seem like one of those odd facts that professors
insist, for their own inscrutable reasons, on having students
memorize. I suggest that instead of memorizing it, you go over the
derivation of that result (eggs and wine) until it makes sense to
you. At that point, you will no longer need to memorize it, since you
will be able to reproduce the result for yourself. It is worth
understanding, and not just for passing economics courses. As we will
see in later chapters, consumer surplus is the essential key to
understanding arguments about policy ("should we have tariffs?") as
well as to figuring out how to maximize profits at Disneyland.

The third result from these chapters is the
equimarginal principle, which tells us that, as a result of our own
rational behavior, the ratio of the marginal utilities of goods is
the same as the ratio of their prices. In addition to helping us
understand consumption, the equimarginal principle in this guise is
one example of a pattern that helps us understand how the high
salaries of physicians are connected to the cost of medical school
and the labors of interning, why we do not get ahead by switching
lanes on the freeway, and how not to make money on the stock
market.

POPCORN-AN APPLICATION

In Chapter 2, I asked why popcorn is sold at a
higher price in movie theaters than elsewhere. While we will not be
ready to discuss possible right answers until Chapter 10, we can at
this point use the idea of consumer surplus to show that the obvious
answer is wrong. The obvious answer is that once the customers are
inside the theater, the owner has a monopoly; by charging them a high
price, he maximizes his profit. What I will show is that far from
maximizing profits, selling popcorn at a high price results in lower
profits than selling popcorn at cost!

To do this, I require the usual economic
assumption that people are rational, plus an important simplifying
assumption--that all consumers are identical. While the latter
assumption is unrealistic, it should not affect the monopoly
argument; if the theater owner charges high prices because he has a
monopoly, he should continue to do so even if the customers are all
the same. Here and elsewhere, the assumption of identical consumers
(and identical producers) very much simplifies our analysis. It is
frequently a good way of getting a first approximation solution to an
economic problem.

The theater owner is selling his customers a
package consisting of the opportunity to watch a film, plus
associated goods such as comfortable seats, clean rest rooms, and the
opportunity to buy popcorn. He charges his customers the highest
price at which he can sell the package. Since the customers are
identical, there is one price that everyone will pay and a slightly
higher price that no one will pay.

In order to decide what to put into the package,
the owner must consider how changes will affect its value to the
customers and hence the maximum he can charge the customers for a
ticket. Suppose, to take a trivial case, he decides to improve the
package by giving every customer a quarter as he comes in the door.
Obviously this will increase the amount the customers are willing to
pay for a ticket by exactly $0.25. The owner is worse off by the time
and trouble spent handing out the coins.

Suppose the theater owner decides that since he
has a monopoly on providing seats in the theater, he might as well
charge $1 for each seat in addition to the admission price. Since
everyone wants a seat, the consumer is paying (say) $4 for an
admission ticket and another $1 for a seat. That is the same as
paying $5 for admission. If the customer is not willing to pay $5 for
the movie, he will be no more willing when the payment is divided
into two pieces; if he is willing to pay $5, the theater owner should
have been charging $5 in the first place.

Now suppose the theater owner is trying to decide
whether to sell popcorn in the theater at $1/carton or not sell it at
all. One advantage to selling popcorn is that he gets money for the
popcorn; another is that customers prefer a theater that sells
popcorn to one that does not and are therefore willing to pay more
for admission. How much more?

Figure 4-8 shows a customer's demand curve for
popcorn. At $1/carton, he buys 1 carton. The shaded area is his
consumer surplus--$0.25. That means (by the definition of consumer
surplus) that the customer is indifferent between being able to buy
popcorn at $1/carton and being unable to buy any popcorn but being
given $0.25; the opportunity to buy popcorn at $1/carton is worth
$0.25 to him. Making the popcorn available at that price is
equivalent to handing each customer a quarter as he walks in the
door; it makes the package offered by the theater (movie plus
amenities--including popcorn) $0.25 more valuable to him, so the
theater owner can raise the admission price by $0.25 without driving
off the customers. The owner should start selling popcorn, provided
that the cost of doing so is less than $1.25/customer. That is what
he gets from selling the popcorn--a dollar paid for the popcorn plus
$0.25 more paid for admission because the opportunity to buy popcorn
is now part of the package.

One theater customer's demand curve for
popcorn. The shaded triangle is the
consumer surpluse frombuying popcorn at $1/carton. The colored region
(ABEDC) is the increase in his consumer surplus if price falls from
$1/carton to $0.50/carton.

Is $1/carton the best price? Assume that, as shown
on Figure 4-8, the marginal
cost to the owner of producing popcorn
(the additional cost for each additional carton produced) is
$0.50/carton. He can produce as many cartons as he likes, at a cost
of $0.50 (for popcorn, butter, wages, and so forth) for each
additional carton. Suppose he lowers the price of popcorn from $1 to
$0.50. He is now selling each customer 2 cartons instead of 1, so his
revenue is still $1/customer. His costs have risen by $0.50/customer,
since he has to produce 2 cartons instead of 1. Consumer surplus,
however, has risen by the colored area on Figure 4-8, which is $0.75;
he can raise the admission price by that amount without losing
customers. His revenue from selling popcorn is unchanged, his costs
have risen by $0.50/ customer, and his revenue from admissions has
risen by $0.75/customer; so his profits have gone up by
$0.25/customer.

The argument is a general one; it does not depend
on the particular numbers I have used. As long as the price of
popcorn is above its marginal cost of production, profit can be
raised by lowering the price of popcorn to marginal cost (MC on
Figure 4-8) and raising the price of admission by the resulting
increase in consumer surplus. The reduction in price reduces the
owner's revenue on the popcorn that he was selling already by its
quantity times the reduction--rectangle ABDC. The cost of producing
the additional popcorn demanded because of the lower price is just
covered by what the consumers pay for it, since the price of a carton
of popcorn is equal to the cost of producing it; on Figure 4-8, both
the additional cost and the additional revenue from selling popcorn
are rectangle DEHG. Consumer surplus goes up by the colored area in
the figure--rectangle ABDC plus triangle BDE. Since the owner can
raise his admission price by the increase in consumer surplus, his
revenue goes up by (ABDC + BDE) (increased admission) + (DEHG - ABDC)
(change in revenue from selling popcorn). His cost goes up by DEHG,
so his profit goes up by the area of triangle BDE.

The same argument can be put in words, without
reference to the diagram: "So far as the popcorn already being sold
is concerned, the price reduction is simply a transfer from the
theater owner to the customer, so revenue from selling popcorn goes
down by the same amount that consumer surplus goes up (ABDC). So far
as the additional popcorn sold at the lower price is concerned, the
customer pays the owner its production cost (DEHG) and is left with
its consumer surplus (BDE). So if we lower the price of popcorn to
its marginal cost, consumer surplus goes up by more than revenue from
popcorn goes down. The theater owner can transfer the consumer
surplus to his own pocket by raising the admission price to the
theater; by doing so (and reducing popcorn to cost), he increases his
profit by the consumer surplus on the additional popcorn
(BDE)."

This shows that any price for popcorn above
production cost lowers the profits of the theater owner, when the
effect of the price of popcorn on what customers are willing to pay
to come to the theater is taken into account.

We are now left with a puzzle. We have used
economics to prove that a theater owner maximizes his profits by
selling popcorn at cost. Economics also tells us that theater owners
should want to maximize their profits and know how to do so. That
implies that they will sell popcorn at cost. Yet they apparently do
not. Something is wrong somewhere; there must be a mistake either in
the logic of the argument, in its assumptions, or in our observation
of what theaters actually do. We will return to that puzzle, and two
possible solutions, in Chapter 10.

OPTIONAL SECTION

CONSUMER SURPLUS AND MEASURING WITH A
(SLIGHTLY) RUBBER RULER

In using the equality between the marginal value
curve and the demand curve to derive a downward-sloping demand curve
earlier in this chapter, I discussed some of the problems of
measuring value in goods instead of in utility. We are now in a
position to see how the same problem affects the concept of consumer
surplus.

Suppose a new good becomes available at price P.
Consumer surplus, the area under the demand curve for the new good
and above a horizontal line at P, is supposed to be the net benefit
to me in dollars of being able to buy the new good--the increase in
my utility divided by my marginal utility for a dollar. But as I
increase my expenditure on the new good, I must be decreasing my
total expenditure on all old goods. The less I spend on something,
the less I consume of it; the less I consume, the greater its
marginal utility. So after I have adjusted my consumption pattern to
include the new good, the marginal utility of all other goods has
risen. Since the marginal utility of a dollar is simply the utility
of what I can buy with it, the marginal utility of a dollar has also
increased. But the original discussion of marginal utility, marginal
value, and consumer surplus treated the marginal utility of a dollar
(usually called the marginal utility of income) as a constant.

The reason this is a good approximation for most
purposes is shown in Figure 4-9. I assume that I am initially
consuming 25 different goods, A-Y, and a twenty-sixth good, Z,
becomes available at a price Pz. The graphs show my marginal
utility for goods A, B, and Z. In the initial situation (shown by the
dashed lines), I am dividing all of my income among goods A-Y in such
a way that the marginal utility of an additional dollar's worth of
each good is the same. The price of good A is assumed to be $1/unit
(the units could be pounds, gallons, or whatever, depending on what
sort of good it is); of B, $2/unit.

After good Z becomes available, I rearrange my
expenditure so that I again have the same marginal utility per dollar
on each unit. Since some of my income is now going to Z, I must be
spending less on each other good, as shown by the solid lines in the
figure. If I simply transferred all of the expenditure away from one
good, its marginal utility per dollar would rise, the marginal
utility per dollar of the other goods would stay the same, and I
would no longer be satisfying the equimarginal principle and hence no
longer maximizing my utility. So instead, I reduce my expenditure a
little on each good, raising the marginal utility of each by the same
amount. The result is that I am now consuming Qa -Qa , of good A, Qb -Qb of good B, and so forth; by the equimarginal principle we
have

MU(Qa- Qa )/Pa = MU(Qb -Qb)/Pb = . . . = MU(Qz)/Pz . (Equation 1)

Since total expenditure is unchanged, the
reduction in expenditure on goods A-Y must equal the new expenditure
on good Z, so

QaPa
+ QbPb + . . . = QzPz. (Equation 2)

Marginal utility curves for three goods,
showing the situation before and after the third good becomes
available. When good Z becomes available,
the consumer buys less of goods A-Y and spends the money on Z
instead. Colored regions show utility losses on goods A and B which
(with similar losses on C-Y, not shown) add up to the colored region
representing expenditure on good Z.

Since I am consuming 25 other goods, the decrease
in consumption of each of them when I start consuming the new good as
well is very small, as shown on the figures. So the marginal utility
of a dollar's worth of the good is almost the same after the change
as before.

To put the derivation of consumer surplus in terms
of utility rather than in dollars (and so make it more precise),
consider the narrow colored areas in Figures 4-9a and 4-9b. They
represent the utility loss as a result of the decreased consumption
of goods A and B. They are almost equal to the narrow rectangles
whose height is MU(Q -Q)and whose
width is Q, where "Q" is Qa in Figure 4-9a and
Qb in Figure
4-9b. If you sum the areas of those rectangles (for all of goods
A-Y), you get

Total area = MU(Qa -Qa)Qa + MU(Qb -Qb)Qb + . . . .

Substituting in from Equation 1 we have

= (MU(Qz)/Pz) x { PaQa+ PbQb + . . . ,

which by Equation 2

= (MU(Qz)/Pz)(PzQz) = MU(Qz)Qz= colored area on Figure
4-9c.

Since the total utility I get from consuming
Qz of Z is
the area under the MU curve (the shaded area plus the colored area)
my net gain is the shaded area--my consumer surplus measured in
utiles.

The one approximation in all of this was ignoring
the part of the narrow rectangles on Figures 4-9a and 4-9b that was
shaded but not colored. That difference becomes smaller, relative to
the colored part, the larger the number of different goods being
consumed; as the number of goods goes to infinity, the ratio of
shaded to colored goes to zero. So consumer surplus as we measure it
(the area under an ordinary demand curve and above price) and
consumer surplus as we define it (the value to the consumer of being
able to buy the good) are equal for a consumer who divides his
expenditure among an infinite number of goods, and are nearly equal
for a real consumer, who divides his expenditure among a large but
finite number of goods.

A mathematical argument is not really satisfactory
unless it can be translated into English. This particular one
translates into a short dialogue:

Query: "When a new good becomes available, you get
consumer surplus by spending money on that good. But do you not lose
the consumer surplus on the other goods you are now not buying with
that money?"

Response: "If you are consuming many goods, you
get the money to buy the new good by giving up a marginal unit of
each of the others: the last orange that was barely worth buying, the
trip you weren't sure you wanted to take. The marginal unit is worth
just what you pay for it--that is why it is marginal--so it generates
no surplus."

4. Figure 4-12 shows your marginal and total
utility curves for persimmons. Are persimmons a good? A bad? Both?
Explain.

5. Figure 4-13a shows your demand curve for Diet
Coke.

a. Approximately how much better off are you being
able to buy all the Diet Coke you want at $5/gallon than not being
able to buy any?

b. How much better off are you being able to buy
all the Diet Coke you want at $3/ gallon than at $5/gallon?

Your demand curves for diet coke and
sapphires. For Problems 5 and 7.

6. Estimate, to within a factor of 10, what
percentage of all water used in the United States is used to drink.
Give your sources. Is the common conception of a "water shortage" as
a situation where people are going thirsty an accurate one? What does
this tell us about the difference between the marginal value of water
at a quantity of a few gallons a week and the marginal value of water
at the quantity we actually consume? (The numerical part of this
cannot be answered from anything in the book; it is intended to give
you practice in the useful art of back-of-the-envelope
calculations--very rough estimates of real-world magnitudes--while at
the same time connecting the abstract examples of the chapter to
something real.)

7. Figure 4-13b shows your demand curve for
sapphires. For religious reasons, sapphires can neither be bought nor
sold. You accidentally discover 100 carats of sapphires. How much
better off are you?

8. Figure 4-14a shows your demand curve for red
tape. There is no market for red tape, but the government, which is
trying to reduce its inventory, orders you to buy 50 pounds of it at
$0.20/pound. How much better or worse off are you as a result?

After you have finished buying and paying for
them, there is an announcement over the store's public address
system; a special Mardi Gras sale has just started, and colored
marshmallows are now only $0.50/bag.

b: Do you buy more? If so, how many?

c. What is your total consumer surplus from buying
marshmallows--including those you bought initially and any others you
bought during the sale?

10. In the example worked out in the text, how
would profit be changed by a further reduction in the price of
popcorn to $0.25/carton?

Your demand curves for red tape and
marshmallows. For Problems 8 and 9.